Geodesic
Stabilization of a~solution to the Cauchy problem for a~nondivergence parabolic equation with growing lower order coefficients
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 97-109

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We obtain sharp sufficient conditions on the growth of lower order coefficients of a second-order parabolic equation under which a solution to the Cauchy problem stabilizes to zero uniformly in $x$ on every compact set $K\in\mathbb R^N$ in some classes of growing initial functions. 
@article{TM_2010_270_a6,
     author = {V. N. Denisov},
     title = {Stabilization of a~solution to the {Cauchy} problem for a~nondivergence parabolic equation with growing lower order coefficients},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {97--109},
     publisher = {mathdoc},
     volume = {270},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2010_270_a6/}
}
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V. N. Denisov. Stabilization of a~solution to the Cauchy problem for a~nondivergence parabolic equation with growing lower order coefficients. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 270 (2010), pp. 97-109. http://geodesic.mathdoc.fr/item/TM_2010_270_a6/